MOTION 85 of a diameter cos 0 = — 1, therefore v\ = 3 v. When they are separated by an angle which is an odd multiple of -, cos 6 = 0; therefore Vi=v \/5. 2i PROBLEMS. 1. An automobile is moving at the rate of 30 miles an hour in a direction at right angles to a train which is making 40 miles an hour; find the velocity of the automobile with respect to the train. 2. Two trains pass each other on parallel tracks, in opposite directions. A passenger in one of the trains observes that it takes the other train 4 seconds to pass him. What is the length of the other train if the velocities of the two trains are 50 and 40 miles per hour? 3. A man of height h walks on a level street away from an electric lamp of height PL If the velocity of the man is v, find the velocity of the end of his shadow (a) with respect to the ground and (b) with respect to the man. 4. Two particles move, in opposite directions, on the circumference of the same circle with the same constant speed. Find an expression for their relative velocity and see what this expression becomes at special positions of the particles. 6. A train is moving due north at the rate of 50 miles an hour. The wind is blowing from the southeast with a velocity of 20 miles an hour. Find the apparent direction and magnitude of the wind to a man on the train. 6. The wind seems to blow from the north to an automobile party traveling westward at the rate of 15 miles an hour. On doubling the speed of the automobile the wind appears to come from the northwest. Find the actual direction and magnitude of the velocity of the wind. 7. Find the velocity of a particle moving on the circumference of a circle with uniform speed relative to another particle moving with equal speed in a diameter of the circle. 8. Express the speed of a mile a minute in the C.G.S. units. 9. Express the C.G.S. unit of velocity in miles per hour. 10. Prove that x2 + y2 = f2 + r202. 11. Prove analytically that vx = vr cos 6 — Vp sin 6, vy = vr sin 6 + vp cos 6. 12. Prove graphically that vr — vx cos Q + vy sin 6, vp = vv cos B — vx sin 6.